I bought one without a serial number and Paul at the shop said it would be okay to make a mark of some kind behind the neck plate, as a protection against theft. Not a bad idea but neck plates like this are a dime a dozen. Paint in some DNA maybe, like they do sometimes with artworks. Verification would be a tad laborious though I got nuthin'.

I would say that yes, it would be legal. In fact, in the case of some things - like custom, made-from-scratch guns, cars, or motorcycles - it's not only legal, it's required. In those cases, something like your initials and a number - ABC 001, for instance - is perfectly acceptable. How you do it is, of course, up to you...